matlab的特殊矩阵

骑猪看日落 2022-04-06 13:12 259阅读 0赞

# matlab的特殊矩阵 #

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**魔方矩阵 magic()  
范德蒙矩阵vander()  
希尔布特矩阵 hilb()  
伴随矩阵—compan§------p为由高阶到低阶的系数  
帕斯卡矩阵----pascal()**

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### 魔方矩阵 magic() ###

行，列，主，副对角线上的元素之和相等

>> A = magic(8)
    A =
        64     2     3    61    60     6     7    57
         9    55    54    12    13    51    50    16
        17    47    46    20    21    43    42    24
        40    26    27    37    36    30    31    33
        32    34    35    29    28    38    39    25
        41    23    22    44    45    19    18    48
        49    15    14    52    53    11    10    56
         8    58    59     5     4    62    63     1
    >> sum(A(1,:))
    ans =
       260
    >> sum(A(:,1))
    ans =
       260

### 范德蒙矩阵vander() ###

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80Mjg3ODc1OA_size_16_color_FFFFFF_t_70]

>> A = vander(1:5)
    A =
        1     1     1     1     1
       16     8     4     2     1
       81    27     9     3     1
      256    64    16     4     1
      625   125    25     5     1

### 希尔布特矩阵 hilb() ###

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80Mjg3ODc1OA_size_16_color_FFFFFF_t_70 1]  
病态矩阵，阶数越大，越严重

>> format rat
    >> H = hilb(4)
    H =
           1              1/2            1/3            1/4     
           1/2            1/3            1/4            1/5     
           1/3            1/4            1/5            1/6     
           1/4            1/5            1/6            1/7

### 伴随矩阵—compan§------p为由高阶到低阶的系数 ###

多项式伴随矩阵的特征值就是多项式的根  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80Mjg3ODc1OA_size_16_color_FFFFFF_t_70 2]

>> P = [1,-2,-5,6]
    P =
         1    -2    -5     6
    >> A = compan(P)
    A =
         2     5    -6
         1     0     0
         0     1     0
    >> p = [1,-2,-5,6]
    >> x = roots(P)--------求根
    x =
      -2.000000000000000
       3.000000000000003
       1.000000000000000
    >> format 
    >> A = eig(A)----------伴随矩阵的特征值
    A =
       -2.0000
        3.0000
        1.0000

### 帕斯卡矩阵----pascal() ###

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80Mjg3ODc1OA_size_16_color_FFFFFF_t_70 3]

>> P= pascal(5)
    P =
         1     1     1     1     1
         1     2     3     4     5
         1     3     6    10    15
         1     4    10    20    35
         1     5    15    35    70
    >> inv(P)
    ans =
         5   -10    10    -5     1
       -10    30   -35    19    -4
        10   -35    46   -27     6
        -5    19   -27    17    -4
         1    -4     6    -4     1

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